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Evaluating the Interpretability of Threshold Operators

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Knowledge Engineering and Knowledge Management (EKAW 2022)

Abstract

Weighted Threshold Operators are n-ary operators that compute a weighted sum of their arguments and verify whether it reaches a certain threshold. They have been extensively studied in the area of circuit complexity theory, as well as in the neural network community under the name of perceptrons. In Knowledge Representation, they have been introduced in the context of standard Description Logics (DL) languages by adding a new concept constructor, the Tooth operator (\(\nabla \!\!\!\nabla \)). Tooth expressions can provide a powerful yet natural tool to represent local explanations of black box classifiers in the context of Explainable AI. In this paper, we present the result of a user study in which we evaluated the interpretability of tooth expressions, and we compared them with Disjunctive Normal Forms (DNF). We evaluated interpretability through accuracy, response time, confidence, and perceived understandability by human users. We expected tooth expressions to be generally more interpretable than DNFs. In line with our hypothesis, the study revealed that tooth expressions are generally faster to use, and that they are perceived as more understandable by users who are less familiar with logic. Our study also showed that the type of task, the type of DNF, and the background of the respondents affect the interpretability of the formalism used to represent explanations .

This research is partially supported by Italian National Research Project PRIN2020 2020SSKZ7R and by unibz RTD2020 project HULA.

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Notes

  1. 1.

    Interpretability describes the possibility to comprehend a black box model and to present the underlying basis for decision-making in a way that is understandable to humans [13].

  2. 2.

    More precisely, non-nested tooth-expressions are not able to represent the XOR. Nested tooth can however overcome this difficulty.

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Acknowledgment

The authors thank Oliver Kutz, Nicolas Troquard, Pietro Galliani, and Antonella De Angeli for taking the pre-test and providing precious feedback about the user study.

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Correspondence to Guendalina Righetti , Daniele Porello or Roberto Confalonieri .

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A A Examples used in the questionnaires

A A Examples used in the questionnaires

  1. 1.
    • DNF1: \( A \sqcup B \)

    • DNF2: \( A \sqcup (\lnot A \sqcap B)\)

    • DNF3: \( (A \sqcap B) \sqcup (\lnot A \sqcap B) \sqcup (A \sqcap \lnot B)\)

    • Tooth: \(\nabla \!\!\!\nabla ^1 ((A,1), (B,1))\)

  2. 2.
    • DNF1: \( (\lnot A \sqcap C) \sqcup B \)

    • DNF2: \( (A \sqcap B) \sqcup (\lnot A \sqcap C) \sqcup (\lnot A \sqcap B \sqcap \lnot C)\)

    • DNF3: \((A \sqcap B \sqcap C) \sqcup (\lnot A \sqcap B \sqcap C) \sqcup (\lnot A \sqcap B \sqcap \lnot C) \sqcup (\lnot A \sqcap \lnot B \sqcap C) \sqcup (A \sqcap B \sqcap \lnot C)\)

    • Tooth: \(\nabla \!\!\!\nabla ^2 ((\lnot A,1), (B,2), (C,1)) \equiv \nabla \!\!\!\nabla ^1 ((A, -1), (B, 2), (C, 1))\)

  3. 3.
    • DNF1: \((\lnot A \sqcap B) \sqcup C\)

    • DNF2: \((\lnot A \sqcap B) \sqcup (A \sqcap \lnot B \sqcap C) \sqcup (A \sqcap B \sqcap C) \sqcup (\lnot A \sqcap \lnot B \sqcap C) \)

    • DNF3: \( (\lnot A \sqcap B \sqcap C) \sqcup (\lnot A \sqcap B \sqcap \lnot C) \sqcup (A \sqcap \lnot B \sqcap C) \sqcup (A \sqcap B \sqcap C) \sqcup (\lnot A \sqcap \lnot B \sqcap C) \)

    • Tooth: \(\nabla \!\!\!\nabla ^2 ((A,-1), (B,2), (C,3))\)

  4. 4.
    • DNF1: \((A \sqcap B) \sqcup (B \sqcap C) \sqcup (A \sqcap C)\)

    • DNF2: \((A \sqcap B) \sqcup (A \sqcap \lnot B \sqcap C) \sqcup (\lnot A \sqcap B \sqcap C)\)

    • DNF3: \((A \sqcap B \sqcap C) \sqcup (\lnot A \sqcap B \sqcap C) \sqcup ( A \sqcap \lnot B \sqcap C) \sqcup ( A \sqcap B \sqcap \lnot C)\)

    • Tooth: \(\nabla \!\!\!\nabla ^2 ((A,1), (B,1), (C,1))\)

  5. 5.
    • DNF1: \((A \sqcap D) \sqcup (A \sqcap B \sqcap C) \sqcup (D \sqcap B)\sqcup (D \sqcap C)\)

    • DNF2: \((A \sqcap D) \sqcup (A \sqcap B \sqcap C \sqcap \lnot D) \sqcup (\lnot A \sqcap B \sqcap D)\sqcup (\lnot A \sqcap \lnot B \sqcap C \sqcap D)\)

    • DNF3: \((\lnot A \sqcap \lnot B \sqcap C \sqcap D) \sqcup (\lnot A \sqcap B \sqcap \lnot C \sqcap D) \sqcup (\lnot A \sqcap B \sqcap C \sqcap D)\sqcup (A \sqcap \lnot B \sqcap \lnot C \sqcap D) \sqcup (A \sqcap \lnot B \sqcap C \sqcap D) \sqcup (A \sqcap B \sqcap \lnot C \sqcap D) \sqcup (A \sqcap B \sqcap C \sqcap \lnot D)\sqcup (A \sqcap B \sqcap C \sqcap D)\)

    • Tooth: \(\nabla \!\!\!\nabla ^5 ((A,3), (B,1), (C,1), (D,4))\)

  6. 6.
    • DNF1: \((A \sqcap B) \sqcup (A \sqcap C) \sqcup (A \sqcap D) \sqcup (B \sqcap D)\)

    • DNF2: \((A \sqcap B \sqcap \lnot D) \sqcup (\lnot A \sqcap B \sqcap C \sqcap D) \sqcup (A \sqcap \lnot B \sqcap C \sqcap \lnot D) \sqcup (\lnot A \sqcap B \sqcap \lnot C \sqcap D) \sqcup (A \sqcap D)\)

    • DNF3: \((\lnot A \sqcap B \sqcap \lnot C \sqcap D) \sqcup (\lnot A \sqcap B \sqcap C \sqcap D) \sqcup (A \sqcap \lnot B \sqcap \lnot C \sqcap D) \sqcup (A \sqcap \lnot B \sqcap C \sqcap \lnot D) \sqcup (A \sqcap \lnot B \sqcap C \sqcap D) \sqcup (A \sqcap B \sqcap \lnot C \sqcap \lnot D) \sqcup (A \sqcap B \sqcap \lnot C \sqcap D) \sqcup (A \sqcap B \sqcap C \sqcap \lnot D) \sqcup (A \sqcap B \sqcap C \sqcap D) \)

    • Tooth: \(\nabla \!\!\!\nabla ^3 ((A,2), (B, 1.5), (C, 1), (D, 1.5)) \)

  7. 7.
    • DNF 1: \((A \sqcap B) \sqcup (A \sqcap C \sqcap D) \sqcup (B \sqcap C \sqcap D)\)

    • DNF 2: \( (A \sqcap B) \sqcup (\lnot A \sqcap B \sqcap C \sqcap D) \sqcup (A \sqcap \lnot B \sqcap C \sqcap D) \)

    • DNF 3: \( (A \sqcap B \sqcap C \sqcap D) \sqcup (A \sqcap B \sqcap \lnot C \sqcap \lnot D) \sqcup (A \sqcap B \sqcap \lnot C \sqcap D) \sqcup (A \sqcap B \sqcap C \sqcap \lnot D) \sqcup (\lnot A \sqcap B \sqcap C \sqcap D) \sqcup (A \sqcap \lnot B \sqcap C \sqcap D) \)

    • Tooth: \(\nabla \!\!\!\nabla ^4 ((A, 2), (B, 2), (C, 1), (D, 1))\)

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Righetti, G., Porello, D., Confalonieri, R. (2022). Evaluating the Interpretability of Threshold Operators. In: Corcho, O., Hollink, L., Kutz, O., Troquard, N., Ekaputra, F.J. (eds) Knowledge Engineering and Knowledge Management. EKAW 2022. Lecture Notes in Computer Science(), vol 13514. Springer, Cham. https://doi.org/10.1007/978-3-031-17105-5_10

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  • DOI: https://doi.org/10.1007/978-3-031-17105-5_10

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